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Theorem iunxsngf 29347
Description: A singleton index picks out an instance of an indexed union's argument. (Contributed by Mario Carneiro, 25-Jun-2016.) (Revised by Thierry Arnoux, 2-May-2020.)
Hypotheses
Ref Expression
iunxsngf.1 𝑥𝐶
iunxsngf.2 (𝑥 = 𝐴𝐵 = 𝐶)
Assertion
Ref Expression
iunxsngf (𝐴𝑉 𝑥 ∈ {𝐴}𝐵 = 𝐶)
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥)   𝑉(𝑥)

Proof of Theorem iunxsngf
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eliun 4515 . . 3 (𝑦 𝑥 ∈ {𝐴}𝐵 ↔ ∃𝑥 ∈ {𝐴}𝑦𝐵)
2 rexsns 4208 . . . 4 (∃𝑥 ∈ {𝐴}𝑦𝐵[𝐴 / 𝑥]𝑦𝐵)
3 iunxsngf.1 . . . . . 6 𝑥𝐶
43nfcri 2756 . . . . 5 𝑥 𝑦𝐶
5 iunxsngf.2 . . . . . 6 (𝑥 = 𝐴𝐵 = 𝐶)
65eleq2d 2685 . . . . 5 (𝑥 = 𝐴 → (𝑦𝐵𝑦𝐶))
74, 6sbciegf 3461 . . . 4 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦𝐵𝑦𝐶))
82, 7syl5bb 272 . . 3 (𝐴𝑉 → (∃𝑥 ∈ {𝐴}𝑦𝐵𝑦𝐶))
91, 8syl5bb 272 . 2 (𝐴𝑉 → (𝑦 𝑥 ∈ {𝐴}𝐵𝑦𝐶))
109eqrdv 2618 1 (𝐴𝑉 𝑥 ∈ {𝐴}𝐵 = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1481  wcel 1988  wnfc 2749  wrex 2910  [wsbc 3429  {csn 4168   ciun 4511
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-v 3197  df-sbc 3430  df-sn 4169  df-iun 4513
This theorem is referenced by:  esum2dlem  30128  fiunelros  30211
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